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Pre-Publicacoes do Departamento de MatematicaUniversidade de CoimbraPreprint Number 04–41

THE N-MEMBRANES PROBLEM FOR QUASILINEARDEGENERATE SYSTEMS

ASSIS AZEVEDO, JOSE-FRANCISCO RODRIGUES AND LISA SANTOS

Abstract: We study the regularity of the solution of the variational inequal-ity for the problem of N -membranes in equilibrium with a degenerate operatorof p-Laplacian type, 1 < p < ∞, for which we obtain the corresponding Lewy-Stampacchia inequalities. By considering the problem as a system coupled throughthe characteristic functions of the sets where at least two membranes are in contact,we analyze the stability of the coincidence sets.

1. Introduction

In an open bounded subset Ω of Rd, d ≥ 1, we consider the quasi-linear

operator

Av = −∇ · a(x,∇v) in D′(Ω),

where a : Ω × Rd −→ R

d is a Caratheodory function, and the followingvariational inequality for the N -membranes problem

(u1, . . . , uN) ∈ KN :N

∑

i=1

∫

Ω

a(x,∇ui) · ∇(vi − ui) ≥N

∑

i=1

∫

Ω

fi(vi − ui), (1)

∀ (v1, . . . , vN) ∈ KN .

Here we shall consider the convex subset KN of the Sobolev space[

W 1,p(Ω)]N

,1 < p <∞, defined by

KN =

(v1, . . . , vN) ∈[

W 1,p(Ω)]N

: v1 ≥ · · · ≥ vN , a. e. in Ω, (2)

vi − ϕi ∈W1,p0 (Ω), i = 1, . . . , N

,

Received December 28, 2004.The second author was partially supported by Project POCTI/MAT/34471/2000 of the Por-

tuguese FCT (Fundacao para a Ciencia e Tecnologia). The third author acknowledges partialsupport from the FCT (Fundacao para a Ciencia e Tecnologia).

1

2 ASSIS AZEVEDO, JOSE-FRANCISCO RODRIGUES AND LISA SANTOS

where we give ϕ1, . . . , ϕN ∈ W 1,p(Ω), such that KN 6= ∅. For instance, if∂Ω ∈ C0,1 is a Lipschitz boundary, it suffices to assume, in the trace sense,that

ϕ1 ≥ · · · ≥ ϕN on ∂Ω.

In (1) we shall assume the N forces given by the functions

f1, . . . , fN ∈ Lq(Ω) ⊂ W−1,p′(Ω) (3)

where W−1,p′(Ω) denotes the dual space of W 1,p0 (Ω), so that p′ = p

p−1 is theconjugate exponent of p and, by Sobolev imbeddings, q = 1 if p > d, q > 1if p = d or q = dp

dp+p−d if 1 0, if ξ 6= η, (6)

for given constants α, β > 0, the general theory of variational inequalities forstrictly monotone operators (see [14], [9]) immediately yields the existenceand uniqueness of solution to the N -membranes problem (1). If we chooseas a model for the N -membranes in equilibrium, each one under the actionof the forces fi and attached to rigid supports at height ϕi, the minimizationfunctional

E(u1, . . . , uN) =N

∑

i=1

∫

Ω

[

1

p|∇ui|

p − fiui

]

in the convex set of admissible displacements given by (2), we obtain thevariational inequality (1) associated with the p-Laplacian

Av = −∆pv = −∇ ·(

|∇v|p−2∇v)

, 1 < p <∞.

The N -membranes problem was considered in [4] for linear elliptic operators,where for differentiable coefficients the regularity of the solution in Sobolevspaces W 2,p(Ω) was shown for p ≥ 2 (hence also in C1,λ(Ω) for 0 < λ =1 − d

p < 1) extending earlier results of [23] for the two membranes problem.

Noting the analogy (and relation) with the one obstacle problem, it wasobserved in those problems that the C2-regularity of the solution cannot beexpected in general, even with very smooth data. Considering the analogyof the two and three membranes problem, respectively with the one and the

THE N -MEMBRANES PROBLEM FOR QUASILINEAR DEGENERATE SYSTEMS 3

two obstacle problems, in [1] we have shown the Lewy-Stampacchia typeinequalities

i∧

j=1

fj ≤ Aui ≤N∨

j=i

fj a.e. in Ω, i = 1, . . . , N, (7)

for general second order linear elliptic operators with measurable coefficientsand, in cases N = 2 and N = 3 we have established sufficient conditions onthe external forces for the stability of the coincidence sets

x ∈ Ω : uj(x) = uj+1(x), j = 1, . . . , N − 1, (8)

where two consecutive membranes touch each other. In (7) we use the nota-tion

k∨

i=1

ξi = ξ1 ∨ . . . ∨ ξk = supξ1, . . . , ξk

andk

∧

i=1

ξi = ξ1 ∧ . . . ∧ ξk = infξ1, . . . , ξk

and we also denote ξ+ = ξ ∨ 0 and ξ− = −(ξ ∧ 0). In order to prove (7)we shall approximate, in Section 2, the solution (u1, . . . , uN) of (1) by solu-tions (uε1, . . . , u

εN) of a suitable system of Dirichlet problems for the operator

A associated to a particular new monotone perturbation that extends thebounded penalization, as ε → 0, of obstacle problems (see [9] or [19] andtheir references). Under the further assumptions on the strong monotonicityof the vector field a(x, ξ) with respect to ξ, i.e., if for some α > 0,

[

a(x, ξ) − a(x, η)]

· (ξ − η) ≥

α|ξ − η|p if p ≥ 2,

α (|ξ| + |η|)p−2 |ξ − η|2 if 1 2, and of the order ε1/2, if 1 0 and on the Lq-norms of f1, . . . , fN .This type of estimate that appears in [20] for the obstacle problem in casep ≥ 2 seems new for 1 < p < 2. The inequalities (7) are a consequenceof the fact that each Aui is a Lq function and we can regard u1 and uNas solutions of one obstacle problems and all the other ui, 1 < i < N , assolutions of two obstacles problems, to which we can apply the well-known

Lewy-Stampacchia inequalities (see, for instance [19], [22], [20] or [17] andtheir references). Another important consequence of these properties is thereduction of the regularity of the solution of the N -membranes problem tothe regularity of each equation

Aui = hi a.e. in Ω, i = 1, . . . , N. (10)

Therefore, in Section 3, we conclude from the well-known properties for weaksolutions of quasilinear elliptic equations (see [11] and [16]) that the solutionsui are in fact Holder continuous, provided q > d

p in (3), or have Holder

continuous gradient (see [5]) if q > dpp−1 and the operator A has the stronger

structural properties, for a.e. x ∈ Ω,

d∑

i,j=1

∂ai

∂ηj(x, η) ξiξj ≥ α0|η|

p−2|ξ|2 (11)

∣

∣

∣

∣

∂ai

∂ηj(x, η)

∣

∣

∣

∣

≤ α1|η|p−2 and

∣

∣

∣

∣

∂ai

∂xj(x, η)

∣

∣

∣

∣

≤ α1|η|p−1 (12)

for some positive constants α0, α1 and all η ∈ Rd \ 0, ξ ∈ R

d and alli, j = 1, . . . , d. We may even conclude that each

ui ∈ C0,λ(Ω) or ui ∈ C1,λ(Ω), i = 1, . . . , N,

provided the Dirichlet data ϕi and ∂Ω have the corresponding required reg-ularity (see Section 3). Finally in Section 4 we study the stability of thecoincidence sets (8) in terms of the convergence of their characteristic func-tions. For this purpose, we define, for a.e. x ∈ Ω and for 1 ≤ j < k ≤ N ,

the following N(N−1)2 coincidence sets

Ij,k = x ∈ Ω : uj(x) = · · · = uk(x) (13)

and notice that the coincidence sets defined in (8) are simply Ij,j+1. Besidesthat, Ij,k = Ij,j+1 ∩ . . . ∩ Ik−1,k. Set

χj,k(x) = χ

Ij,k(x) =

1 if uj(x) = · · · = uk(x)

0 otherwise .(14)

In [1] we have shown that the solution (u1, u2, u3) of (1) for N = 3 with alinear operator in fact satisfies a.e. in Ω

Au1 = f1 + 12(f2 − f1)χ1,2 + 1

6(2f3 − f2 − f1)χ1,3

Au2 = f2 − 12(f2 − f1)χ1,2 + 1

2(f3 − f2)χ2,3 + 16(2f2 − f1 − f3)χ1,3

Au3 = f3 − 12(f3 − f2)χ2,3 + 1

6(2f1 − f2 − f3)χ1,3

(15)

which extends the remark of [24] for the case N = 2 that corresponds to thefirst two equations of (15) with χ2,3 ≡ 0 (and consequently also χ1,3 ≡ 0). As

f1 6= f2 a.e. in Ω

is a sufficient condition for the convergence of the unique coincidence set I1,2in case N = 2, additionally

f2 6= f3, f1 6=f2 + f3

2, f3 6=

f1 + f2

2a.e. in Ω

in case N = 3 are sufficient conditions for the convergence of the threecoincidence sets I1,2, I2,3 and I1,3, with respect to the perturbation of theforces f1, f2, f3 (see [1] for a direct proof). In section 4 we extend to arbitraryN the system (15) by showing that, for given forces (f1, . . . , fN) the solution(u1, . . . , uN) of (1) solves a system of the form

Aui = fi +∑

1≤j<k≤N, j≤i≤k

bj,ki [f ]χj,k a.e. in Ω, i = 1, . . . , N, (16)

where, in (16), each bj,ki [f ] represents a certain linear combination of the

forces. We denote the average of fj, . . . , fk by

〈f〉j,k =fj + · · · + fk

k − j + 1, 1 ≤ j ≤ k ≤ N, (17)

and we shall establish the following assumption on the averages of the forces

〈f〉i,j 6= 〈f〉j+1,k a.e. in Ω, for all i, j, k ∈ 1, . . . , N with i ≤ j < k, (18)

as a sufficient condition for the stability of the coincidence sets Ij,k in theN -membranes problem.

2. Approximation by bounded penalization

In this section we approximate the variational inequality using a boundedpenalization. Defining

ξ0 = max

f1+···+fi

i : i = 1, . . . , N

ξi = i ξ0 − (f1 + · · · + fi), for i = 1, . . . , N ,

(19)

we observe that

ξi ≥ 0 if i ≥ 1

(ξi−1 − ξi−2) − (ξi − ξi−1) = fi − fi−1 if i ≥ 2.(20)

For ε > 0, let θε be defined as follows:

θε : R −→ R

s 7→

0 if s ≥ 0

sε if −ε < s < 0

−1 if s ≤ −ε.

(21)

The approximated problem is given by the system

Auεi + ξiθε(uεi − uεi+1) − ξi−1θε(u

εi−1 − uεi ) = fi in Ω,

uεi |∂Ω= ϕi, i = 1, · · · , N,

(22)

with the convention uε0 = +∞, uεN+1 = −∞.

Proposition 2.1. If the operator A satisfies the assumptions (4), (5) and

(6), problem (22) has a unique solution (uε1, . . . , uεN) ∈

[

W 1,p(Ω)]N

. Thissolution satisfies

uεi ≤ uεi−1 + ε, for i = 2, . . . , N. (23)

Proof : Existence and uniqueness of solution of the problem (22) is an imme-diate consequence of the theory of strictly monotone and coercive operators(see [14]). In fact, summing the N equations of the system, each one multi-plied by a test function wi, problem (22) implies that

N∑

i=1

∫

Ω

〈Auεi , wi〉+〈Bv,w〉 =N

∑

i=1

∫

Ω

fiwi, ∀w = (w1, . . . , wN) ∈[

W 1,p(Ω)]N,


where

〈Bv,w〉 =N

∑

i=1

∫

Ω

(

ξiθε(vi − vi+1) − ξi−1θε(vi−1 − vi))

wi

with the same convention vε0 = +∞, vεN+1 = −∞, satisfies

〈Bv −Bw, v − w〉 =

N−1∑

i=1

∫

Ω

ξi

(

θε(vi − vi+1) − θε(wi − wi+1))

(

(vi − vi+1) − (wi − wi+1))

≥ 0,

since ξi ≥ 0, for i = 1, . . . , N and θε is monotone nondecreasing.To prove (23), multiplying the i−th equation of (22) by (uεi − uεi−1 − ε)+

and integrating on Ω, noticing that (uεi − uεi−1 − ε)+|∂Ω

= 0 we obtain,∫

Ω

Auεi (uεi − uεi−1 − ε)+ =∫

Ω

[fi − ξiθε(uεi − uεi+1) + ξi−1θε(u

εi−1 − uεi )] (u

εi − uεi−1 − ε)+

=

∫

Ω

[fi − ξiθε(uεi − uεi+1) − ξi−1] (u

εi − uεi−1 − ε)+

since θε(uεi−1−u

εi )(u

εi −u

εi−1−ε)

+ = −(uεi −uεi−1−ε)

+. In particular, becauseθε(u

εi − uεi+1) ≥ −1, we have

∫

Ω

Auεi (uεi − uεi−1 − ε)+ ≤

∫

Ω

[fi + ξi − ξi−1] (uεi − uεi−1 − ε)+. (24)

With similar arguments, if we multiply, for i ≥ 2, the ( i− 1)−th equation of(22) by (uεi − uεi−1 − ε)+ and integrate on Ω we obtain,

∫

Ω

Auεi−1(uεi − uεi−1 − ε)+ ≥

∫

Ω

[fi−1 + ξi−1 − ξi−2] (uεi − uεi−1 − ε)+. (25)

From inequalities (24) and (25) we have, using (20),∫

Ω

(

a(x,∇uεi ) − a(x,∇uεi−1))

· ∇(uεi − uεi−1 − ε)+ =∫

Ω

(Auεi − Auεi−1) (uεi − uεi−1 − ε)+

≤

∫

Ω

[

fi − fi−1 +(

ξi − ξi−1

)

−(

ξi−1 − ξi−2

)]

(uεi − uεi−1 − ε)+ = 0.


From the strict monotony (6) of a, it follows that uεi ≤ uεi−1 + ε a.e. in Ω.

Proposition 2.2. If (uε1, . . . , uεN) and (u1, . . . , uN) are respectively the solu-

tion of the problem (22) and the solution of the problem (1) then

(uε1, . . . , uεN) −− (u1, . . . , uN) in

[

W 1,p(Ω)]N

-weak, when ε→ 0.

Proof : Multiplying the i-th equation of (1) by vi−uεi , where (v1, . . . , vN) ∈ K

and uε = (uε1, . . . , uεN), integrating over Ω and summing, we obtain

N∑

i=1

∫

Ω

a(x, uεi ) · ∇(vi − uεi ) + 〈Buε, v − uε〉 =N

∑

i=1

∫

Ω

fi(vi − uεi ).

Noticing that 〈Bv, v − uε〉 = 0 and due to the monotonicity of the operatorB proved in (24),

N∑

i=1

∫

Ω

a(x,∇uεi ) · ∇(vi − uεi ) ≥N

∑

i=1

∫

Ω

fi(vi − uεi ) (26)

and using (6) we conclude that

N∑

i=1

∫

Ω

a(x,∇vi) · ∇(vi − uεi ) ≥N

∑

i=1

∫

Ω

fi(vi − uεi ). (27)

From (4) and (5) we easily deduce the uniform boundedeness of (uε1, . . . , uεN)ε

in[

W 1,p(Ω)]N

. So, there exists (u∗1, . . . , u∗N) ∈

[

W 1,p(Ω)]N

such that

(uε1, . . . , uεN) −− (u∗1, . . . , u

∗N) in

[

W 1,p(Ω)]N

-weak, when ε→ 0

and, letting ε→ 0 in (27) we obtain

N∑

i=1

∫

Ω

a(x,∇vi) · ∇(vi − u∗i ) ≥N

∑

i=1

∫

Ω

fi(vi − u∗i ) ∀ (v1, . . . , vN) ∈ K.

Besides that, using (23), u∗1 ≥ · · · ≥ u∗n. Since we also have u∗i |∂Ω = ϕi, for i =

1, . . . , N , then (u∗1, . . . , u∗N) ∈ K. The hemicontinuity of the operator A allows

us to conclude that (u∗1, . . . , u∗N) actually solves the variational inequality

(1) and the uniqueness of solution of the variational inequality implies thatu∗i = ui, i = 1, . . . , N .

We present now two lemmas that will be used to prove the next theorem.The first lemma states that, under certain circunstancies, weak convergenceimplies strong convergence. The second lemma is a reverse type Holderinequality.

Lemma 2.3. ([3], p. 190) Under the assumptions (4), (5) and (6), whenε→ 0, if

uε − u −− 0 in W1,p0 (Ω) (28)

and∫

Ω

[a(x,∇uε) − a(x,∇u)] · ∇(uε − u) −→ 0 (29)

then

uε − u −→ 0 in W1,p0 (Ω)-strong.

Lemma 2.4. ([21], p. 8) Let 0 < r < 1 and r′ = rr−1. If F ∈ Lr(Ω),

FG ∈ L1(Ω) and

∫

Ω

|G(x)|r′

dx <∞ in a bounded domain Ω of Rd, then one

has(

∫

Ω

|F (x)|rdx

)1

r

≤

(∫

Ω

|F (x)G(x)|dx

)(∫

Ω

|G(x)|r′

dx

)− 1

r′

. (30)

Theorem 2.5. Let (uε1, . . . , uεN) and (u1, . . . , uN) denote, respectively, the

solutions of the problems (22) and (1). Under the assumptions (4), (5) and(6),

i) (uε1, . . . , uεN) −−→

ε→0(u1, . . . , uN) in

[

W 1,p(Ω)]N

.

ii) If, in addition, a is strongly monotone, i.e., satisfies (9), then thereexists a positive constant C, independent of ε, such that, for all i =1, . . . , N ,

‖∇(uεi − ui)‖Lp(Ω) ≤

Cε1

p if p ≥ 2,

Cε1

2 if 1 < p ≤ 2.

Proof : i) Choose, for i = 1, . . . , N , vi =N∨

k=i

uεk in (1). Indeed, since vi−1 ≥ vi

a.e. in Ω and vi − ϕi ∈W1,p0 (Ω), (v1, . . . , vN) ∈ K and we have

N∑

i=1

∫

Ω

a(x,∇ui) · ∇

(

N∨

k=i

uεk − ui

)

≥N

∑

i=1

∫

Ω

fi

(

N∨

k=i

uεk − ui

)

.


So,

N∑

i=1

∫

Ω

a(x,∇ui) · ∇ (uεi − ui) ≥N

∑

i=1

∫

Ω

fi (uεi − ui)

+N

∑

i=1

∫

Ω

a(x,∇ui) · ∇

(

uεi −N∨

k=i

uεk

)

+N

∑

i=1

∫

Ω

fi

(

N∨

k=i

uεk − uεi

)

.

On the other hand, by (26),

N∑

i=1

∫

Ω

a(x,∇uεi ) · ∇(uεi − ui) ≤N

∑

i=1

∫

Ω

fi(uεi − ui),

and we conclude thatN

∑

i=1

∫

Ω

[

a(x,∇uεi ) − a(x,∇ui)]

· ∇(uεi − ui) ≤

N∑

i=1

∫

Ω

a(x,∇ui) · ∇

(

N∨

k=i

uεk − uεi

)

−N

∑

i=1

∫

Ω

fi

(

N∨

k=i

uεk − uεi

)

=N

∑

i=1

∫

Ω

(

Aui − fi)

(

N∨

k=i

uεk − uεi

)

.

(31)

Here we have used the fact that Aui ∈ Lq(Ω), for i = 1, . . . , N , since we knowthat

fi − ξi−1 ≤ Auεi = −ξiθε(uεi − uεi+1) + ξi−1θε(u

εi−1 − uεi ) + fi ≤ fi + ξi,

by (22) and −1 ≤ θε ≤ 0. Noticing that, from (23),

0 ≤N∨

k=i

uεk − uεi ≤ uεi + (N − i+ 1)ε− uεi ≤ (N − i+ 1) ε (32)

it is immediate to conclude that

0 ≤N

∑

i=1

∫

Ω

[


· ∇(uεi − ui) ≤ Cε. (33)

and, since (28) and (29) hold, then, by Lemma 2.3, for each i = 1, . . . , N ,

uεi −→ ui when ε→ 0 in W 1,p(Ω).

ii) From (33) and using the strong monotonicity of the operator a, we have

• if p ≥ 2,

α

N∑

i=1

∫

Ω

|∇(uεi − ui)|p ≤

N∑

i=1

∫

Ω

[


· ∇(uεi − ui) ≤ Cε,

• if 1 < p < 2, using also the strong monotonicity of the operator a and(33),

α

N∑

i=1

∫

Ω

(

|∇uε| + |∇ui|)p−2

|∇(uεi − ui)|2 ≤

N∑

i=1

∫

Ω

[


· ∇(uεi − ui) ≤ Cε,

(34)

Let Ωi = x ∈ Ω : |∇uεi | + |∇ui| 6= 0. We may use the reverseinequality (30) with r = p

2 , noticing that 0 < r < 1 and r′ = pp−2 ,

setting F = |∇(uεi − ui)|2 and G =

(

|∇uεi | + |∇ui|)p−2

. Then weobtain, for i = 1, . . . , N ,

(∫

Ωi

|∇(uεi − ui)|p

)2

p

dx ≤

(∫

Ωi

|∇(uεi − ui)|2(

|∇uεi | + |∇ui|)p−2

dx

) (∫

Ωi

(

|∇uεi | + |∇ui|)pdx

)2−p

p

.

Since by (34)∫

Ωi

|∇(uεi − ui)|2(

|∇uεi | + |∇ui|)p−2

dx ≤1

αCε

and

∃Mp > 0 :

(∫

Ωi

(

|∇uεi | + |∇ui|)pdx

)2−p

p

≤ Mp,

the conclusion follows immediately summing the N inequalities above.

3. Lewy-Stampacchia inequalities and regularity

As a consequence of the approximation by bounded penalization we knowalready that each

Aui ∈ Lq(Ω), i = 1, . . . , N

and so, we can use the analogy with the obstacle problem to show furtherregularity of the solution ui. In [12] Lewy and Stampacchia have shownthat the solution of the obstacle problem for the Laplacian satisfies a dualinequality, which in fact holds in more general cases, as it was observed in[7] or [2] for nonlinear operators. Summarizing the known results for theone and the two obstacles problem that we shall apply to the N -membranesproblem, the following theorem may be proved as in [19] or [17].

Theorem 3.1. Given ϕ, ψ1, ψ2 ∈W 1,p(Ω), (1 d, q > 1 if p = d orq = dp

dp+p−d if 1 < p < d) such that

Kψ1

ψ2=

v ∈ W 1,p(Ω) : ψ1 ≥ v ≥ ψ2 a.e. in Ω, v − ϕ ∈ W1,p0 (Ω)

6= ∅, (35)

the unique solution to the variational inequality

u ∈ Kψ1

ψ2:

∫

Ω

a(x,∇u) · ∇(v − u) ≥

∫

Ω

f(v − u), ∀v ∈ Kψ1

ψ2(36)

under the assumptions (4), (5) and (6) satisfies the Lewy-Stampacchia in-equality

f ∧ Aψ1 ≤ Au ≤ f ∨ Aψ2 a.e. in Ω. (37)

Remark 3.2. Setting ξ1 = (Aψ1 − f)− and ξ2 = (Aψ2 − f)+ and using thepenalization function θε of the previous section we may approach, as ε → 0,the solution of (36) by the solutions uε of the equation

Auε + ξ2θε (uε − ψ2) − ξ1θε (ψ1 − uε) = f in Ω (38)

with the Dirichlet boundary condition uε = ϕ on ∂Ω. Noting that

f ∧ Aψ1 = f − (Aψ1 − f)− and f ∨ Aψ2 = f + (Aψ2 − f)+

we easily conclude (37) from the analogous inequalities that are satisfied foreach uε.


Remark 3.3. We may consider that Theorem 3.1, although stated for twoobstacles problem, also contains the case of only one obstacle. Indeed, bytaking ψ1 ≡ +∞, (36) is a lower obstacle problem and (37) reads

f ≤ Au ≤ f ∨ Aψ2, for u ≥ ψ2, a.e. in Ω (39)

and by taking ψ2 ≡ −∞, (36) is an upper obstacle problem for which (37)reads

f ∧ Aψ1 ≤ Au ≤ f, for u ≤ ψ1, a.e. in Ω. (40)

Remark 3.4. In [17], for more general operators and under a strong mono-tonicity assumption of the type (9), which however is not necessary in ourTheorem 3.1, it was shown that each inequality of (37) still holds indepen-dently of each other in the duality sense, provided Aψ1 − f and/or Aψ2 − f

are in V ∗p′ =

[

W−1,p′(Ω)]+

−[

W−1,p′(Ω)]+

, i.e., in the ordered dual space of

W1,p0 (Ω).

Theorem 3.5. The solution (u1, . . . , uN) of the N-membranes problem, un-der the assumptions (4), (5) and (6), satisfies the following Lewy-Stampacchiatype inequalities

f1 ≤ Au1 ≤ f1 ∨ · · · ∨ fNf1 ∧ f2 ≤ Au2 ≤ f2 ∨ · · · ∨ fN

...f1 ∧ · · · ∧ fN−1 ≤ AuN−1 ≤ fN−1 ∨ fNf1 ∧ · · · ∧ fN ≤ AuN ≤ fN

a.e. in Ω (41)

Proof : Observe that choosing (v, u2, . . . , uN) ∈ KN , with v ∈ Ku2, we see

that u1 ∈ Ku2(as in (35) with ψ1 = +∞) solves the variational inequality

(36) with f = f1, and so by (39) we have

f1 ≤ Au1 ≤ f1 ∨ Au2 a.e. in Ω.

Analogously, we see that uj ∈ Kuj−1

uj+1solves the two obstacles problem (36)

with f = fj, j = 2, . . . , N − 1, and satisfies, by (37),

fj ∧ Auj−1 ≤ Auj ≤ fj ∨ Auj+1 a.e. in Ω.

Since uN ∈ KuN−1, by (40), also satisfies

fN ∧ AuN−1 ≤ AuN ≤ fN a.e. in Ω,

(41) is easily obtained by simple iteration.

Although for p > d, the Sobolev inclusion W 1,p(Ω) ⊂ C0,λ(Ω) for 0 < λ =1 − d

p < 1, immediately implies the Holder continuity of the solutions ui ofthe N -membranes problem, this property still holds for 1 < p ≤ d by usingthe fact that each Aui is in the same Lq(Ω) as the forces fi, i = 1, . . . , N .So under the classical assumptions of [11] (see also [16]) we may state forcompleteness the following regularity result.

Corollary 3.6. Under the assumptions (3)-(6) for 1 dp in

(3), the solution (u1, . . . , uN) of (1) is such that,

ui ∈ C0,λ(Ω) for some 0 < λ < 1, i = 1, . . . , N

and is also in C0,λ(Ω) if, in addition, each ϕi ∈ C0,λ(∂Ω) and ∂Ω is smooth,for instance, of class C0,1.

Remark 3.7. The above classical result for equations was also shown to holdfor the one obstacle problem, for instance, in [6] or in [15], or for the twoobstacles problems in [10], under more general assumptions on the data. Itwould be interesting to obtain the Holder continuity of the solution of (1) di-rectly under the classical and more general assumptions of each fi ∈W−1,s(Ω)for s > d

p−1.

A more interesting regularity is the Holder continuity of the gradient ofthe solution, by analogy with the results for solutions of degenerate ellipticequations. For instance, as a consequence of the inequalities (41) and theresults of [5] on the C1,λ local regularity of weak solutions, as well as in theregularity up to the boundary of [13], we may also state the following results.

Corollary 3.8. Under the stronger differentiability properties (11), (12), if(3) holds with q > dp

p−1, the solution (u1, . . . , uN) of (1) is such that

ui ∈ C1,λ(Ω) for some 0 < λ < 1, i = 1, . . . , N

and is also in C1,λ(Ω) if, in addition, each ϕi ∈ C1,γ(∂Ω), for some γ (λ ≤γ < 1) and fi ∈ L∞(Ω) for all i = 1, . . . , N .

For differentiable strongly coercive vector fields satisfying the assumptions(11), (12), with p = 2, there is no degeneration of the operator A and strongerregularity in W 2,s(Ω) may be obtained also from the fact that (41) holds forthe solution of the N -membranes problem. For instance, as in Theorem 3.3of [9], page 114 (see also Remark 4.5 of [19], page 244), we may prove thefollowing result.

Corollary 3.9. Let (11), (12) hold for p = 2, suppose ∂Ω ∈ C1,1 and eachfi ∈ L∞(Ω), ϕi ∈ W 2,∞(Ω), i = 1, . . . , N . Then the solution (u1, . . . , uN) of(1) is such that

ui ∈ W 2,s(Ω) ∩ C1,γ(Ω), i = 1, . . . , N, for all 1 ≤ s <∞ and 0 ≤ γ < 1.(42)

Remark 3.10. For N linear operators of the form

aki (x, ξ) =d

∑

j=1

akij(x)ξj k = 1, . . . , N,

the regularity (42) was shown in [4] for every s ≥ 2 and, with the sameoperators with lower order terms in [1] for s > 1 if d = 2 and s ≥ 2d

d+2 ifd ≥ 3. For the case of two membranes with linear operators, earlier resultswere shown in [23], by using similar regularity results for the one obstacleproblem. In spite of this analogy, the optimal W 2,∞ regularity of solutions toobstacle problems is an open problem for the N-membranes system.

Remark 3.11. In the case of two membranes with constant mean curvature,i.e., when A is the minimal surface operator and f1 and f2 are constants in a

smooth domain with mean curvature H∂Ω of ∂Ω larger or equal to |f1|∨|f2|d−1 , in

[24] it was shown the existence of a unique solution with the regularity (42).The N-membranes problem for the minimal surface operator, in general, isan open problem.

4. The convergence of the coincidence sets

In this section we prove that, if (un1 , . . . , unN) is the solution of the N -

membranes problem, under the assumptions (4), (5) and (6) with given data

(fn1 , . . . , fnN), n ∈ N, if (fn1 , . . . , f

nN) converges in [Lq(Ω)]N to (f1, . . . , fN),

we have the stability result in Ls(Ω), 1 ≤ s < ∞, for the correspondingcoincidence sets

χun

k=···=unl

−−→n

χuk=···=ul, for 1 ≤ k < l ≤ N.

We begin presenting a lemma that will be needed.

Lemma 4.1. [20] Given functions u, v ∈ W 1,p(Ω), 1 < p < ∞, such thatAu,Av ∈ L1(Ω), we have

Au = Av a.e. in x ∈ Ω : u(x) = v(x).

In what follows we continue using the convention u0 = +∞ and uN+1 =−∞. Given 1 ≤ j ≤ k ≤ N , we define the following sets

Θj,k = x ∈ Ω : uj−1(x) > uj(x) = · · · = uk(x) > uk+1(x). (43)

The first part of the following proposition identifies the value of Aui a.e. oneach coincidence set Ij,k defined in (13). The second part states a necessarycondition satisfied by the forces in order to exist contact among consecutivemembranes.

Proposition 4.2. If j, k ∈ N are such that 1 ≤ j ≤ k ≤ N , we have

i) Aui =

〈f〉j,k a.e. in Θj,k if i ∈ j, . . . , k

fi a.e. in Θj,k if i 6∈ j, . . . , k

ii) if j < k then for all i ∈ j, . . . , k 〈f〉i+1,k ≥ 〈f〉j,i a.e. in Θj,k.

Proof : i) Suppose i ∈ j, . . . , k (the other case has a similar and simplerproof). For a.e. x ∈ Θj,k we have uj−1(x) − uj(x) = α > 0 and uk(x) −uk+1(x) = β > 0, for some α = α(x), and β = β(x). Since x belongs to theopen set y ∈ Ω : uj−1(y)−uj(y)−

α2 > 0∩y ∈ Ω : uk(y)−uk+1(y)−

β2 > 0,

there exists δ > 0 such that, for all ϕ ∈ D(B(x, δ)), there exists ε0 > 0 suchthat, if 0 < ε < ε0, then uj−1 ≥ uj ± εϕ and uk ≥ uk+1 ± εϕ. Choose for testfunctions

vr =

ur if r 6∈ j, . . . , k

ur ± εϕ if r ∈ j, . . . , k.

Then

±εk

∑

r=j

∫

Ω

a(x,∇ur) · ∇ϕ ≥ ±εk

∑

r=j

∫

Ω

fr ϕ, ∀ϕ ∈ D(B(x, δ))

andk

∑

r=j

∫

Ω

a(x,∇ur) · ∇ϕ =k

∑

r=j

∫

Ω

fr ϕ, ∀ϕ ∈ D(B(x, δ)).

So we conclude thatk

∑

r=j

Aur =k

∑

r=j

fr, a.e. in B(x, δ).


We know that Aui ∈ L1(Ω), for all i = 1, . . . , N . So, using Lemma 4.1, wehave

Auj = · · · = Aui = · · · = Auk in Θj,k

and we conclude then that

(k − j + 1)Aui = fj + · · · + fk a.e. in Θj,k.

ii) The proof of this item is analogous to the previous one. We choose fortest functions

vr =

ur if r 6∈ j, . . . , i

ur + εϕ if r ∈ j, . . . , i

with ϕ ∈ D(B(x, δ)), ϕ ≥ 0, ε > 0 such that (v1, . . . , vN) ∈ K. We concludethen that

i∑

r=j

∫

Ω

a(x,∇ur) · ∇ϕ ≥i

∑

j=r

∫

Ω

fr ϕ, ∀ϕ ∈ D(B(x, δ)), ϕ ≥ 0,

and so, we have Aui ≥ 〈f〉j,i a.e. in Θj,k. Then using the first part of thisproposition we conclude that

〈f〉j,k ≥ 〈f〉j,i a.e. in Θj,k

or equivalently, that

〈f〉i+1,k ≥ 〈f〉j,i a.e. in Θj,k.

Our goal is to determine a system of N equations, coupled by the charac-

teristic functions of the N(N−1)2 coincidence sets, which is equivalent to the

problem (1). This was done in [23] for the case N = 2 and in [1] for the caseN = 3. The system for N = 2 is simply

Au1 = f1 + f2−f12χu1=u2

Au2 = f2 −f2−f1

2χu1=u2

and for N = 3 is the system (15). From these two examples we see that thedetermination of the coefficients of this system is not a very simple problemof combinatorics. We present the result for the case N in Theorem 4.5.

Definition 4.3. Given f1, . . . , fN ∈ Lq(Ω) we define, for j, k, i ∈ 1, . . . , N,with j < k and j ≤ i ≤ k,

bj,ki [f ] =

〈f〉j,k − 〈f〉j,k−1 if i = j

〈f〉j,k − 〈f〉j+1,k if i = k

2(k−j)(k−j+1)

(

〈f〉j+1,k−1 −12(fj + fk)

)

if j < i < k.

Observe that, if j < i < k, then bj,ki [f ] does not depends on i. It is also

not difficult to see thatk

∑

i=j

bj,ki [f ] = 0. We notice first some auxiliary results

concerning the coefficients bj,ki [f ] that will be needed. From now on we drop

the dependence of bj,ki [f ] on f .

Lemma 4.4.

i) If j ≤ l < r then

r∑

k=l+1

bj,kj =

r − l

r − j + 1

(

〈f〉l+1,r − 〈f〉j,l)

;

In particularr

∑

k=l+1

bj,kj is positive if and only if the average of fl+1, . . . , fr

is greater or equal then the average of fj, . . . , fl.ii) If m < i then

∀ r ∈ i, . . . , Nr

∑

k=i

bm,ki = bm,rr .


Proof : i) We have

r∑

k=l+1

bj,kj =

r∑

k=l+1

(

〈f〉j,k − 〈f〉j,k−1

)

= 〈f〉j,r − 〈f〉j,l

=fj + · · · + fr

r − j + 1−fj + · · · + fl

l − j + 1

=fj + · · · + fl

r − j + 1+fl+1 + · · · + fr

r − j + 1−fj + · · · + fl

l − j + 1

=fl+1 + · · · + fr

r − j + 1−

(r − l)(fj + · · · + fl)

(r − j + 1)(l − j + 1)

=r − l

r − j + 1

(

fl+1 + · · · + fr

r − l−fj + · · · + fl

l − j + 1

)

=r − l

r − j + 1

(

〈f〉l+1,r − 〈f〉j,l)

.

ii) We prove the equality by induction over r. If r = i, the equality is trivial.For r > i

r+1∑

k=i

bm,ki =

r∑

k=i

bm,ki + b

m,r+1i

= bm,rr + bm,r+1i , by induction hypothesis,

= 〈f〉m,r − 〈f〉m+1,r +2

(r −m+ 1)(r −m+ 2)

(

〈f〉m+1,r −1

2

(

fm + fr+1

)

)

=fm + · · · + fr

r −m+ 1−fm+1 + · · · + fr

r −m+

2(fm+1 + · · · + fr)

(r −m)(r −m+ 1)(r −m+ 2)−

fm + fr+1

(r −m+ 1)(r −m+ 2).

Thenr+1∑

k=i

bm,ki =

(

1

r −m+ 1−

1

(r −m+ 1)(r −m+ 2)

)

fm −1

(r −m+ 1)(r −m+ 2)fr+1 +

(

1

r −m+ 1−

1

r −m+

2

(r −m)(r −m+ 1)(r −m+ 2)

)

(

fm+1 + · · · + fr

)

=fm

r −m+ 2−

fr+1

(r −m+ 1)(r −m+ 2)−

fm+1 + · · · + fr

(r −m+ 1)(r −m+ 2)

=fm + · · · + fr+1

r −m+ 2−fm+1 + · · · + fr+1

r −m+ 1

= bm,r+1

r+1 .

We are now able to deduce the system of equations involving the charac-teristic functions of the coincidence sets which is equivalent to the problem(1).

Theorem 4.5.

Aui = fi +∑

1≤j<k≤N, j≤i≤k

bj,kiχj,k a.e. in Ω. (44)

Proof : We prove that the equality is valid a.e. in Θm,r for m, r such that 1 ≤

m ≤ r ≤ N . This is enough because⋃

1≤m≤r≤N

Θm,r = Ω. If i 6∈ m, . . . , r,

then (44) results immediately from Proposition 4.2– i). Supposing then thati ∈ m, . . . , r, using Lemma 4.2, the equality (44), for x ∈ Θm,r becomes

fi +∑

m≤j<k≤r, j≤i≤k

bj,ki = 〈f〉m,r.

We prove now this equality by induction on i−m.If i−m = 0, then

fi +∑


bj,ki = fm +

∑

m<k≤r

bm,km

= fm +∑

m<k≤r

(

〈f〉m,k − 〈f〉m,k−1

)

= 〈f〉m,r.

By the induction step, if i−m > 0, then

fi +∑


bj,ki = fi +

∑

m+1≤j<k≤r, j≤i≤r

bj,ki +

∑

i≤k≤r

bm,ki

= 〈f〉m+1,r +r

∑

k=i

bm,ki by induction hypothesis

= 〈f〉m+1,r + bm,rr by Lemma 4.4–ii)

= 〈f〉m,r.

We have now the main result of this section.

Theorem 4.6. Given n ∈ N, let (un1 , . . . , unN) denote the solution of problem

(1) with given data (fn1 , . . . , fnN) ∈ [Lq(Ω)]N , with q as in (3). Suppose that

fni −−→n

fi in Lq(Ω), i = 1, . . . , N. (45)

Then

uni −−→n

ui in W 1,p(Ω), i = 1, . . . , N. (46)

If, in addition, the limit forces satisfy

〈f〉i,j 6= 〈f〉j+1,k, for all i, j, k ∈ 1, . . . , N with i ≤ j < k, (47)

then, for any 1 ≤ s <∞,

∀ j, k ∈ 1, . . . , N, j < k χun

j =···=unk

−−→n

χuj=···=uk in Ls(Ω). (48)

Before proving the theorem we need another auxiliary lemma:

Lemma 4.7. Let n ∈ N and a1, . . . , an ∈ R be such thatn

∑

r=j

ar > 0 for all

j = 1, . . . , n. Then the inequality

a1 Y1 + · · · + an Yn ≤ 0,

with the restrictions 0 ≤ Y1 ≤ · · · ≤ Yn, has only the trivial solution Y1 =· · · = Yn = 0.

Proof : If n = 1 the conclusion is immediate. Supposing the result proved forn, let us prove it for n+ 1:

0 ≥ a1 Y1 + · · · + an Yn + an+1 Yn+1

≥ a1 Y1 + · · · + an Yn + an+1 Yn

since Yn+1 ≥ Yn ≥ 0 and an+1 > 0. Then

0 ≥ a1 Y1 + · · · + (an + an+1) Yn

and, because the result is true for n, then Y1 = · · · = Yn = 0 and, therefore,since an+1 > 0, we also have Yn+1 = 0.

Proof of Theorem 4.6: The convergence (46) of the solutions is an immediateconsequence of a theorem due to Mosco. For simplicity, we write χuj=···=uk =χj,k and we denote χun

j =···=unk

by χnj,k Let j, k ∈ 1, . . . , N with j < k. Since

0 ≤ χj,k ≤ 1, there exists χ∗

j,k ∈ Lq(Ω) such that(

χnj,k

)

n∈Nconverges to χ∗

j,k

in Lq(Ω)-weak. Of course we have

0 ≤ χ∗j,k ≤ 1, because 0 ≤ χn

j,k ≤ 1

χ∗m,r ≤ χ∗

j,k (if m ≤ j < k ≤ r), because χnm,r ≤ χnj,k.

(49)

Besides that, letting n→ ∞ in the equality χnj,k(

unj − unk)+

≡ 0, we conclude

χ∗j,k (uj − uk)

+ = 0 a.e. in Ω. (50)

Consider now the system (44), with the coefficients b substituted by bn, fordata fn1 , . . . , f

nN , with n ∈ N,

Auni = fni +∑

j<k≤N, j≤i≤k

(

bn)j,k

iχnj,k a.e. in Ω, i = 1, . . . , N.

Passing to the weak limit in Lq(Ω), when n→ ∞, we have

Aui = fi +∑

j<k≤N, j≤i≤k

bj,kiχ∗j,k a.e. in Ω, i = 1, . . . , N.

Subtracting the equality (44) for the limit solution from this one, we obtain∑

j<k≤N, j≤i≤k

bj,ki

(

χj,k − χ∗

j,k

)

= 0 a.e. in Ω, i = 1, . . . , N. (51)

For k > j, let Yj,k denote χj,k − χ∗j,k. To complete the proof we only need to

show that, for j < k, Yj,k ≡ 0, i.e.,(

χnj,k

)

n∈Nconverges to χj,k in Lq(Ω)-weak.

From equation (50) we know that

∀ j < k Yj,k ≡ 0 in uj 6= uk = uj > uk. (52)

Fix j0 and k0 such that j0 < k0. Using (52), we only need to see thatYj0,k0

≡ 0 in Ij0,k0= uj0 = · · · = uk0

. It is enough then to prove it in twocases: i) Θj0,r, for r ≥ j0; ii) Θm,r for m < j0 and r ≥ k0.

In the first case, using (52), we have Yj,k ≡ 0 in Θj0,r, if j < j0 or k > r.So, letting i = j0 on equation (51), we have, in Θj0,r,

0 =∑

j<k≤N, j≤j0≤k

bj,kj0Yj,k

=∑

j0≤j<k≤N, j≤j0≤k≤r

bj,kj0Yj,k

=r

∑

k=j0+1

bj0,kj0

Yj0,k.

We can apply now Lemma 4.7 to conclude that Yj0,k = 0 in Θj0,r for k ∈j0 + 1, . . . , r, since

• for x ∈ Θj0,r, Yj0,r(x) = 1−χ∗j0,k

(x) and, using (49), Yj0,j0+1(x) ≤ · · · ≤Yj0,r(x);

• for l ≥ j0, by Lemma 4.4–i),

r∑

k=l+1

bj0,kj0

=r − l

r − j0 + 1

(

〈f〉l+1,r − 〈f〉j0,l)

,

which is positive, by Lemma 4.2–ii), as x ∈ Θj0,r, and (47).

In the second case, in Θm,r (m < j0 and r ≥ k0)

0 ≤ Yj0,k0= χ

j0,k0− χ∗

j0,k0

= 1 − χ∗j0,k0

since m < j0 < k0 ≤ r

≤ 1 − χ∗m,k0

by (49)

= χm,k0

− χ∗m,k0

= Ym,k0

= 0 as in the previous case.

Notice that, since χj0,k0is a characteristic function,

(

χnj0,k0

)

n∈Nconverges in

fact to χj0,k0in Ls(Ω)-strong, for all 1 ≤ s <∞.


Remark 4.8. Notice that, arguing as in Theorem 2.5, under the strongmonotonicity assumption (9), it is easy to show the following continuousdependence result on the data,

N∑

i=1

‖uni − ui‖W 1,p0 (Ω ≤ Cq

N∑

i=1

‖fni − fi‖Lq(Ω),

for q defined as in (3). However, a corresponding L1 estimate for the char-acteristic functions of the coincidence sets, similar to the obstacle problem([19], [20]) seems more difficult to obtain.

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Assis AzevedoDepartment of Mathematics, University of Minho, Campus de Gualtar, 4710–057 Braga,Portugal

E-mail address: [email protected]

Jose-Francisco RodriguesCMUC/University of Coimbra & University of Lisbon/CMAF, Av. Prof. Gama Pinto, 2,1649–003 Lisbon, Portugal


Lisa SantosCMAF/University of Lisbon & Department of Mathematics, University of Minho, Campusde Gualtar, 4710–057 Braga, Portugal


1.Introduction - COnnecting REpositories · 1.Introduction In an open bounded subset Ω of Rd, d≥...

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